dispersive shallow water wave modelling. part iv: numerical simulation ... · dispersive shallow...

Commun. Comput. Phys.doi: 10.4208/cicp.OA-2016-0179d

Vol. 23, No. 2, pp. 361-407February 2018

Dispersive Shallow Water Wave Modelling. Part IV:

Numerical Simulation on a Globally Spherical

Geometry

Gayaz Khakimzyanov1, Denys Dutykh2,∗ and Oleg Gusev1

1 Institute of Computational Technologies, Siberian Branch of the Russian Academyof Sciences, Novosibirsk 630090, Russia.2 LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique,73376 Le Bourget-du-Lac Cedex, France.

Received 23 October 2016; Accepted (in revised version) 26 June 2017

Abstract. In the present manuscript we consider the problem of dispersive wave sim-ulation on a rotating globally spherical geometry. In this Part IV we focus on numer-ical aspects while the model derivation was described in Part III. The algorithm wepropose is based on the splitting approach. Namely, equations are decomposed on auniform elliptic equation for the dispersive pressure component and a hyperbolic partof shallow water equations (on a sphere) with source terms. This algorithm is imple-mented as a two-step predictor-corrector scheme. On every step we solve separatelyelliptic and hyperbolic problems. Then, the performance of this algorithm is illustratedon model idealized situations with even bottom, where we estimate the influence ofsphericity and rotation effects on dispersive wave propagation. The dispersive effectsare quantified depending on the propagation distance over the sphere and on the lin-ear extent of generation region. Finally, the numerical method is applied to a coupleof real-world events. Namely, we undertake simulations of the BULGARIAN 2007 andCHILEAN 2010 tsunamis. Whenever the data is available, our computational resultsare confronted with real measurements.

AMS subject classifications: 76B15, 76B25

PACS: 47.35.Bb, 47.35.FgKey words: Finite volumes, splitting method, nonlinear dispersive waves, spherical geometry,rotating sphere, Coriolis force.

1 Introduction

Until recently, the modelling of long wave propagation on large scales has been per-formed in the framework of Nonlinear Shallow Water Equations (NSWE) implemented

∗Corresponding author. Email addresses: Khaki t.ns .ru (G. Khakimzyanov),Denys.Dutykhuniv-savoie.fr (D. Dutykh), gusev_oleg_igormail.ru (O. Gusev)

http://www.global-sci.com/ 361 c©2018 Global-Science Press

362 G. Khakimzyanov, D. Dutykh and O. Gusev / Commun. Comput. Phys., 23 (2018), pp. 361-407

under various software packages [52]. This model is hydrostatic and non-dispersive [79].Among popular packages we can mention, for example, the TUNAMI code [40] basedon a conservative finite difference leap-frog scheme on real bathymetries. This code hasbeen extensively used for tsunami wave modeling by various groups (see e.g. [91]). Thecode MOST uses the directional splitting approach [84, 85] and is also widely used forthe simulation of tsunami wave propagation and run-up [82,88]. The MGC package [75]is based on a modified MACCORMACK finite difference scheme [24], which discretizesNSWE in spherical coordinates. Obviously, the MGC code can also work in CARTE-SIAN coordinates as well. This package was used to simulate the wave run-up on a real-world beach [37] and tsunami wave generation by underwater landslides [4]. Recentlythe VOLNA code was developed using modern second order finite volume schemes onunstructured grids [22]. Nowadays this code is essentially used for the quantification ofuncertainties of the tsunami risk [3].

All numerical models described above assume the wave to be non-dispersive. How-ever, in the presence of wave components with higher frequencies (or equivalently shorterwavelengths), the frequency dispersion effects may influence the wave propagation. Evenin 1982 in [60] it was pointed out:

[ . . . ] the considerations and estimates for actual tsunamis indicate that nonlinearityand dispersion can appreciably affect the tsunami wave propagation at large distances.

Later this conclusion was reasserted in [68] as well. The catastrophic Sumatra event in2004 [80] along with subsequent events brought a lot of new data all around the globeand also from satellites [54]. The detailed analysis of this data allowed to understandbetter which models and algorithms should be applied at various stages of a tsunami lifecycle to achieve the desired accuracy [12, 62]. The main conclusion can be summarizedas follows: for a complete and satisfactory description of a tsunami wave life cycle onglobal scales, one has to use a nonlinear dispersive wave model with moving (in the gen-eration region [18]) realistic bathymetry. For trans-oceanic tsunami propagation one hasto include also Earth’s sphericity and rotation effects. A whole class of suitable mathe-matical models combining all these features was presented in the previous Part III [43] ofthe present series of papers.

At the present time we have a rather limited amount of published research literaturedevoted to numerical issues of long wave propagation in a spherical ocean. In manyworks (see e.g. [27,39]) Earth’s sphericity is not taken explicitly into account. Instead, theauthors project Earth’s surface (or at least a sub-region) on a tangent plane to Earth insome point and computations are then performed on a flat space using a BOUSSINESQ-type (Weakly Nonlinear and Weakly Dispersive — WNWD) model without taking intoaccount the CORIOLIS force. We notice that some geometric defects are unavoidable inthis approach. However, even in this simplified framework the importance of dispersiveeffects has been demonstrated by comparing the resulting wave field with hydrostatic(NSWE) computations.

In [57] the authors studied the transoceanic propagation of a hypothetical tsunami

G. Khakimzyanov, D. Dutykh and O. Gusev / Commun. Comput. Phys., 23 (2018), pp. 361-407 363

generated by an eventual giant landslide which may take place at La Palma island, whichthe most north-westerly island of the Canary Islands, Spain. Similarly the authors em-ployed a WNWD model, but this time written in spherical coordinates with Earth’s ro-tation effects. However, the employed model could handle only static bottoms. As aresult, the initial fields were generated using a different hydrodynamic model and, then,transferred into the WNWD model as the initial condition to compute the wave long timeevolution. However, when waves approach the shore, another limitation of weakly non-linear models becomes apparent — in coastal regions nonlinear effects grow quickly and,thus, the computations should be stopped before the wave reaches the coast. Otherwise,the numerical results may loose their validity. In [57] it was also shown that the wavedispersion may play a significant rôle on the resulting wave field. Namely, NSWE pre-dict the first significant wave hitting the shore, while WNWD equations predict ratheran undular bore in which the first wave amplitude is not necessarily the highest [58]. Ofcourse, these undular bores cannot be described in the framework of NSWE [33, 69].

An even more detailed study of tsunami dispersion was undertaken recently [28],where also a WNWD model was used, but the dispersion effect was estimated for severalreal-world events. The authors came to interesting ‘uncertain’ conclusions:

[ . . . ] However, undular bores, which are not included in shallow-water theory, mayevolve during shoaling. Even though such bores may double the wave height locally,their effect on inundation is more uncertain because the individual crests are shortand may be strongly affected by dissipation due to wave breaking.

It was also noted that near coasts WNWD model provides unsatisfactory results, thatis why fully nonlinear dispersive models should be employed to model all stages fromtsunami generation to the inundation. The same year a fully nonlinear dispersive modelon a sphere including CORIOLIS effect was derived in [51]. However, in contrast to an-other paper from the same group [31], the horizontal velocity variable is taken as a traceof 3D fluid velocity on a certain surface laying between the bottom and free surface. Theproposed model may have some drawbacks. First of all, the well-posed character of theCAUCHY problem is not clear. A very similar (and much simpler) NWOGU’s model [64]is known to possess instabilities for certain configurations of the bottom [56]. We un-derline also that the authors of [51] did not present so far any numerical simulationswith their fully nonlinear model. A study of dispersive and CORIOLIS effects were per-formed in the WNWD counterpart of their fully nonlinear equations. In order to solvenumerically their spherical BOUSSINESQ-type system a CARTESIAN TVD scheme previ-ously described in [73] was generalized to spherical coordinates. This numerical modelwas implemented as a part of well-known FUNWAVE(-TVD) code [74].

A fully nonlinear weakly dispersive model on a sphere with the depth-averaged ve-locity variable was first derived in [25]. Later it was shown in [26] that the same modelcan be derived without using the potential flow assumption. Moreover, this model ad-mits an elegant conservative structure with the mass, momentum and energy conserva-tions. In particular, the energy conservation allows to control the amount of numerical


viscosity in simulations. The same conservative structure can be preserved while de-riving judiciously weakly nonlinear models as well. Only the expressions of the kineticenergy and various fluxes vary from one model to another. In this way one may obtainthe whole hierarchy of simplified shallow water models on a sphere enjoying the sameformal conservative structure [76]. In particular, it allows to develop a unique numeri-cal algorithm, which can be applied to all models in this hierarchy by changing only thefluxes and source terms in the numerical code.

In this study we develop a numerical algorithm to model shallow water wave prop-agation on a rotating sphere in the framework of a fully nonlinear weakly dispersivemodel, which will be described in the following Section. For numerical illustrations weconsider first model problems on the perfect sphere (i.e. the bottom is even). In thisway we assess the influence of dispersive, sphericity and rotation effects depending onthe propagation distance and on the size of the wave generation region. These methodsare implemented in NLDSW_SPHERE code which is used to produce numerical resultsreported below.

The present manuscript is organized as follows. The governing equations that wetackle in our study are set in Section 2. The numerical algorithm is described in Section 3.Several numerical illustrations are described in Section 4. Namely, we start with testsover a perfect rotating sphere in Section 4.1. Then, as an illustration of medium scalewave propagation we study the BULGARIAN 2007 tsunami in Section 4.2. On large trans-oceanic scales we simulate the 2010 CHILEAN tsunami in Section 4.3. Finally in Section 5we outline the main conclusions and perspectives of our study. Some further details onthe derivation of the non-hydrostatic pressure equation are provided in Appendix A.

2 Problem formulation

The detailed derivation of the fully nonlinear model considered in the present study canbe found in the previous Part III [43]. Here we only repeat the governing equations:

(HRsinθ)t +[Hu

]

λ+

[Hvsinθ

]

θ= 0, (2.1)

(HuRsinθ)t +[

Hu2 + gH2

2

]

λ

+[Huvsinθ

]

θ

= gHhλ −Huvcosθ − HvRsinθ + ℘λ − hλ , (2.2)

(HvRsinθ)t +[Huv

]

λ+

[(Hv2 + g

H2

2

)sinθ

]

θ

= gHhθ sinθ + gH2

2cosθ + Hu2 cosθ + HuRsinθ +

(℘θ − hθ

)sinθ , (2.3)

where R is the radius of a virtual sphere rotating with a constant angular velocity Ω

around the axis Oz of a fixed CARTESIAN coordinate system Oxyz. The origin O of this


coordinate system is chosen so that the plane Oxy coincides with sphere’s equatorialplane.

In order to describe conveniently the fluid flow we choose also a spherical coordinatesystem Oλθr whose origin is located at sphere’s center and it rotates with the sphere.Here λ is the longitude increasing in the rotation direction starting from a certain merid-

ian (0 6 λ < 2π). The other angle θdef:= π

2 − ϕ is the complementary latitude (−π2 < ϕ < π

2 ).Finally, r is the radial distance from sphere’s center. The NEWTONIAN gravity force† actson fluid particles and its vector g is directed towards virtual sphere’s center. The total

water depth Hdef:= η + h > 0 is supposed to be small comparing to sphere’s radius, i.e.

H ≪ R. That is why we can suppose that the gravity acceleration gdef:= |g | and fluid den-

sity ρ are constants throughout the fluid layer. The functions h(λ,θ, t) (the bottom profile)and η(λ,θ, t) (the free surface excursion) are given as deviations from the still water levelη00(θ), which is not spherical due to the rotation effect [43].

By u and v we denote the linear components of the velocity vector:

udef:= Ru1 sinθ , v

def:= Ru2 ,

where u1 = λ and u2 = θ . The CORIOLIS parameter def:= 2Ω cosθ is expressed through

the complementary latitude θ and additionally we can assume that

θ0 6 θ 6 π − θ0 , (2.4)

where θ0 = const ≪ 1 is a small angle. In other words, the poles are excluded fromour computations. In practice, it is not a serious limitations since on the Earth polesare covered with ice and no free surface flow takes place there. The quantities ℘ and are dispersive components of the depth-integrated pressure P and fluid pressure at thebottom p respectively:

P =gH2

2− ℘ , p = gH − .

These dispersive components ℘ and can be computed according to the following for-mulas [43]:

℘ =H3

3R1 +

H2

2R2 , =

H2

2R1 + HR2 , (2.5)

where

R1def:= D(∇·u) − (∇·u)2 , R2

def:= D2 h, u

def:=

(u1,u2).

†Here we understand the force per unit mass, i.e. the acceleration.


To complete model presentation we remind also the definitions of various operators inspherical coordinates that we use:

Ddef:= ∂t + u·∇, ∇

def:=

(∂λ,∂θ

), u·∇ ≡ u1∂λ + u2∂θ ,

∇·u ≡ u1λ+

1J

(Ju2)

θ, J

def:= −R2 sinθ .

In the most detailed form, functions R1,2 can be equivalently rewritten as

R1 ≡ (∇·u)t +1

Rsinθ

[

u(∇·u)λ + v(∇·u)θ sinθ]

− (∇·u)2 ,

R2 ≡ (Dh)t +1

Rsinθ

[

u(Dh)λ + v(Dh)θ sinθ]

,

with

∇·u ≡1

Rsinθ

[

uλ + (vsinθ)θ

]

,

Dh ≡ ht +1

Rsinθ

[

uhλ + vhθ sinθ]

.

The model (2.1)-(2.3) is referred to as ‘fully nonlinear’ one since it was derived with-out any simplifying assumptions on the wave amplitude [43]. In other words, all non-linear (but weakly dispersive) terms are kept in this model. So far we shall refer to thismodel as Fully Nonlinear Weakly Dispersive (FNWD) model. The FNWD model shouldbe employed to simulate water wave propagation in coastal and even in slightly deeperregions over uneven bottoms. Weak dispersive effects in the FNWD model ensure thatwe shall obtain more accurate results than with simple NSWE. Moreover, the FNWDmodel contains the terms coming from moving bottom effects [42]. Consequently, wecan model also the wave generation process by fast or slow bottom motions [18], thus,allowing to model tsunami waves from their generation until the coasts [22]. In this waywe extend the validity region of existing WNWD models [28, 57, 58] by including thewave generation regions along with the coasts where the nonlinearity becomes critical.

Concerning the linear dispersive properties, it is generally believed that models withthe depth-averaged velocity can be further improved in the sense of BONA–SMITH [6]and NWOGU [64]. However, nonlinear dispersive wave models tweaked in this way (seee.g. [59]) have a clear advantage only in the linear one-dimensional (1D) situations. Fornonlinear 3D computations (especially involving the moving bottom [17]) the advantageof ‘improved’ models becomes more obscure comparing to dispersive wave models withthe depth-averaged velocity adopted in our study [9, 77]. Moreover, the mathematicalmodel after such transformations (or ‘improvements’ as they are called in the literature)often looses the energy conservation and GALILEAN’s invariance properties. A successfulattempt in this direction was achieved only recently [11].


Eqs. (2.1)-(2.3) admit also a non-conservative form:

Ht +1

Rsinθ

[

(Hu)λ + (Hvsinθ)θ

]

= 0, (2.6)

ut +1

Rsinθuuλ +

1R

vuθ +g

Rsinθηλ =

1Rsinθ

℘λ − hλ

H−

uv

Rcotθ − v, (2.7)

vt +1

Rsinθuvλ +

1R

vvθ +g

Rηθ =

℘θ − hθ

RH+

u2

Rcotθ + u. (2.8)

In the numerical algorithm presented below we use both conservative and non-conservativeforms for our convenience.

If in conservative (2.1)-(2.3) or non-conservative (2.6)-(2.8) governing equations weneglect dispersive contributions, i.e. ℘ 0, 0, then we recover NSWE on a rotatingattracting sphere [8]:

Ht +1

Rsinθ

[

(Hu)λ + (Hvsinθ)θ

]

= 0,

ut +1

Rsinθuuλ +

1R

vuθ +g

Rsinθηλ = −

uv

Rcotθ − v,

vt +1

Rsinθuvλ +

1R

vvθ +g

Rηθ =

u2

Rcotθ + u.

In order to obtain a well-posed problem, we have to prescribe initially the free surfacedeviation from its equilibrium position along with the initial velocity field. Moreover, theappropriate boundary conditions have to be prescribed as well. The curvilinear coast-line is approximated by a family of closed polygons and on edges e we prescribe the wallboundary condition:

u·n∣∣

e= 0, (2.9)

where n is an exterior normal to edge e. In this way, we replace the wave run-up problemby wave/wall interaction, where a solid wall is located along the shoreline on a certainprescribed depth hw .

3 Numerical algorithm

The system of equations (2.1)-(2.3) is not of CAUCHY–KOVALEVSKAYA’s type, since mo-mentum balance equations (2.2), (2.3) involve mixed derivatives with respect to time andspace of the velocity components u, v. We already encountered this difficulty in the glob-ally flat case [46]. A direct (e.g. finite difference) approximation of governing equationswould lead to a complex system of fully coupled nonlinear algebraic equations in a veryhigh dimensional space. In the globally flat case [44] it was found out that it is more ju-dicious to perform a preliminary decoupling‡ of the system into a scalar elliptic equation

‡We underline that this decoupling procedure involves no approximation and the resulting system of equa-tions is completely equivalent to the base model (2.1)-(2.3).


and a system of hyperbolic equations with source terms [46]. In the present work werealize the same idea for FNWD equations (2.1)-(2.3) on a rotating attracting sphere. Sim-ilarly we shall rewrite the governing equations as a scalar elliptic equation to determinethe dispersive component of the depth-integrated pressure℘ and a hyperbolic system ofshallow water type equations with some additional source terms. With this splitting wecan apply the most appropriate numerical methods for elliptic and hyperbolic problemscorrespondingly.

The derivation of the elliptic equation for non-hydrostatic pressure component℘ canbe found in Appendix A. Here we provide only the final result:

[1

sinθ

℘λ

H−

∇℘·∇h

HΥhλ

]

λ

+

[℘θ

H−

∇℘·∇h

HΥhθ

sinθ

]

θ

− K℘ = F , (3.1)

where

Kdef:= K00 +

∂K01

∂λ+

∂K02

∂θ,

with

K00def:= R2 12(Υ − 3)

H3 Υsinθ , K01

def:=

6hλ

H2Υsinθ, K02

def:=

6hθ

H2Υsinθ .

Here we introduced a new variable Υ defined as

Υdef:= 4 + ∇h·∇h ≡ 4 + |∇h|2 .

The scalar product ∇℘·∇h can be easily expressed as

∇℘·∇h ≡1

R2

℘λhλ

sin2 θ+ ℘θ hθ

.

The right hand side F is defined as

Fdef:=

[1

sinθ

gηλ +Q

Υhλ − Λ1

︸︷︷︸

def=: F1

]

λ

+

[

gηθ +Q

Υhθ − Λ2

sinθ︸︷︷︸

def=: F2

]

θ

− R2 6Q

HΥsinθ +

2sinθ

uλ + (vsinθ)θ

2

− 2(uλvθ − vλuθ

)− 2(uv)λcotθ −

(v2 cosθ

)

θ,

where

Qdef:=

(Λ − g∇η

)·∇h +

1R2 sinθ

u2

sinθhλλ + 2uvhλθ + v2hθθ sinθ

+ htt + 2 u

Rsinθhλt +

v

Rhθ t

︸︷︷︸

def=: B

,


with vector Λdef:= ⊤

(Λ1,Λ2

)whose components are

Λ1def:= −

(2uvcotθ + vR

)sinθ , Λ2

def:= u2 cotθ + uR.

The term B contains all the terms coming from bottom motion effects. If the bottom isstationary, then B ≡ 0.

The particularity of Eq. (3.1) is that it does not contain time derivatives of dynamicvariables H, u and v. This equation is very similar to the elliptic equation derived inthe globally flat case [46]. The differences consist only in terms coming from Earth’ssphericity and rotation effects. It is not difficult to show that under total water depthpositivity assumption H > 0 and condition (2.4), Eq. (3.1) is uniformly elliptic. In orderto have the uniqueness result of the DIRICHLET problem for (3.1), the coefficient K has tobe positive defined [55], i.e. K > 0. In the case of an even bottom (i.e. h ≡ h0 = const)the positivity condition takes the following form:

K =12R2 sinθ

H3 Υ

[

Υ − 3 −Ω

2

2g

Hsin2θ − ηθ sin(2θ) +8H

Υcos(2θ)

]

> 0, (3.2)

where this time the variable Υ is defined simply as

Υdef:= 4 +

Ω4 R2

4g2 sin2(2θ).

Using the values of parameters for our planet, i.e.

R = 6.38×106m, Ω = 7.29×10−5

s−1 , g = 9.81

m

s2 ,

and assuming (2.4) along with the fact that the gradient of the free surface elevation η isbounded, one can show that K ≫ 1. Thus, the condition (3.2) is trivially verified. In thiscase one can construct a finite difference operator with positive definite (grid-)operator.Theoretically, it is not excluded that for large bottom variations locally the coefficient Kmight become negative. In such cases, the conditioning of the discrete system is wors-ened and more iterations are needed to achieve the desired accuracy in solving Eq. (3.1).In the globally flat case the analysis of these cases was performed in [46]. In practice,sometimes it is possible to avoid such complications by applying a prior smoothing op-erator to the bathymetry function h(λ,θ, t).

3.1 Numerical scheme construction

The finite difference counterpart of Eq. (3.1) can be obtained using the so-called integro-interpolation method [49]. In this Section we construct a second order approximation tobe able to work on coarser grids for a fixed desired accuracy.


S B

CD

5 2

3

4

0

j1

j2

1

6

78

A

E

N

W

(a)

A B

CD

1 3

7

0

j1

j2

48

(b)

S B

D

5 2

3

4

0

j1

j2

1

6

8

A

E

N

W

(c)

Figure 1: Integration ontours and nite dieren e sten ils for Eq. (3.1) in an internal (a), boundary (b) and

orner ( ) nodes.

In spherical coordinates we consider a uniform grid with spacings ∆λ and ∆θ alongthe axes Oλ and Oθ correspondingly. In order to derive a numerical scheme, first weintegrate Eq. (3.1) over a rectangle ABCD depicted in Fig. 1(a), with vertices A, B, C andD being the geometrical centers of adjacent cells. After applying GREEN’s formula to theresulting integral, we obtain:

“

BCF

1dθ −

“

ADF

1dθ +

“

DCF

2dλ −

“

ABF

2dλ

−

„

ABCDK℘dλdθ =

„

ABCDFdλdθ . (3.3)

From the double integral on the right hand side we can extract a contour part as well:„

ABCD∇·Fdλdθ =

“

BCF1dθ −

“

ADF1dθ +

“

DCF2dλ −

“

ABF2dλ,

where ∇· ~(·)def:= (·1)λ + (·2)θ is the ‘flat’ divergence operator, vectors F

def:=

(F1, F2

),

(F 1,F 2

)with components defined as

F1 =

1sinθ

℘λ

H−

∇℘·∇h

HΥhλ

, F2 =

℘θ

H−

∇℘·∇h

HΥhθ

sinθ ,

F1 =1

sinθ

gηλ +Q

Υhλ − Λ1

, F2 =

gηθ +Q

Υhθ − Λ2

sinθ .

In order to compute approximatively the integrals, we use the trapezoidal numericalquadrature rule along with a second order finite difference approximation of the deriva-tives in cell centers [34]. As a result one can obtain a finite difference approximation forEq. (3.1) with nine points stencil in every internal node of the grid.


In an analogous way one can construct finite difference approximations in boundarynodes adjacent to a non-permeable wall. Let us assume that the node x j1 , j2 =

(λ j1 ,θ j2

)

belongs to the boundary Γ lying along a parallel represented with a bold solid line inFig. 1(b). The interior of the computational domain lies in North-ward direction to thisparallel. If nodes x j1−1, j2 and x j1+1, j2 also belong to Γ, then, the integration contour ABCDis chosen such that vertices C and D coincide with adjacent cell centers (sharing the com-mon node x j1, j2) and vertices A, B coincide with centers of edges belonging to Γ (seeagain Fig. 1(b)). In this case the rectangle side AB belongs completely to the boundaryΓ. That is why boundary conditions for the function ℘ have to be used while computingthe integrals in the integro-differential equation (3.3).

3.1.1 Boundary conditions treatment

In the present study we consider only the case of wall boundary conditions. In the situa-tion depicted in Fig. 1(b) the boundary condition (2.9) becomes simply

v∣∣

Γ≡ 0.

Thanks to Eq. (2.8), we obtain the following boundary condition for the variable ℘ :

1R

℘θ − hθ

H− gηθ

+u2

Rcotθ + u = 0, x ∈ Γ. (3.4)

In Appendix A it is shown that the dispersive component of the fluid pressure at thebottom is related to other quantities H, u, v and ℘ as

=1Υ

6℘H

+ HQ + ∇℘·∇h

. (3.5)

After substituting the last formula into Eq. (3.4), the required boundary condition takesthe form:

1R

(℘θ

H−

∇℘·∇h

HΥhθ

)

−6hθ

RH2Υ℘∣∣∣∣

Γ

=

1R

(

gηθ +Q

Υhθ

)

−u2

Rcotθ − u

∣∣∣∣

Γ

,

or by using some notations introduced above we can simply write a more compact form:F

2 − K02℘ − F2∣∣

AB= 0. (3.6)

The last boundary condition is used while approximating the integrals over the rectangleside AB. Consider the double integral in the left hand side of the integro-differentialequation (3.3). It can be approximated as„

ABCDK℘dλdθ ≈

„

ABCDK00℘dλdθ

+

“

BCK01dθ −

“

ADK01dθ +

“

DCK02dλ −

“

ABK02dλ

℘(0),


and after introducing the following approximations:“

ABF

2dλ ≈ F2(0)∆λ ,

“

ABK02dλ ≈ K02(0)∆λ ,

“

ABF2dλ ≈ F2(0)∆λ ,

we come to an important conclusion: all the terms coming from integrals›

AB vanish thanksto the boundary condition (3.6). Consequently, the implementation of boundary conditionsturns out to be trivial in the integro-interpolation method employed in our study. Forboundary cells of other types, we use the same method of integro-interpolating approxi-mations to obtain difference equations. As an illustration, in Fig. 1(c) we depict anotherone of the eight possible configurations of angular nodes. Since we write a differenceequation for every grid node (interior and boundary), the total number of equations co-incides with the total number of discretization points.

3.1.2 Computational miscellanea

In the most general case a realistic computational domain for the wave propagation hasa complex shape. Generally it is not convex and due to the existence of islands it mightbe multiply connected. It has some implications for linear solvers that we can use tosolve the difference equations described above. First of all, due to the large scale na-ture of problems considered in this study, we privilege iterative schemes for the sake ofcomputational efficiency. Then, due to geometrical and topological reasons describedabove, we adopt a simple but efficient method of Successive Over–Relaxation (SOR) [90].This method contains a free parameter , which can be used to accelerate the conver-gence. The optimal value of ⋆ , which ensures the fastest convergence is in general un-known. This question was studied theoretically for the POISSON equation with DIRICH-LET boundary conditions in a rectangle [71]. So, in this case the optimal value of ⋆ wasshown to belong to the interval (1,2). For example, if the mesh is taken uniform in eachside ℓ of a square with the spacing h = ℓ

N , then

⋆ =2

1 + sin πhℓ

.

If we take ℓ = 1 and N = 30 we obtain the optimal value ⋆ ≈ 1.81. Another observationis that in the limit N → +∞ the optimal value ⋆ → 2. In our numerical experiments weobserved the same tendency: with the mesh refinement the optimal relaxation parameter⋆ for the discretized equation (3.1) approaches 2 as well. In practice, we took the values⋆ ∈ [1.85,1.95] depending on the degree of refinement.

For given functions ℘ and the system of conservative equations (2.1)-(2.3) is ofhyperbolic type under the conditions (2.4) and water depth positivity H(x, t) > 0, ∀t > 0.Thanks to this property we have in our disposal the whole arsenal of numerical tools thathave been developed for hyperbolic systems of equations [29,30]. In the one-dimensionalcase we opted for predictor–corrector schemes [46,50] with a free scheme parameter θn

j1 , j2.

A judicious choice of this parameter ensures the TVD property and the monotonicity of


solutions at least for scalar equations [47]. Some aspects of predictor–corrector schemesin two spatial dimensions are described in [78]. In the present study we employ thepredictor–corrector scheme with θn

j1 , j2≡ 0, which minimizes the numerical dissipation

(and makes the scheme somehow more fragile). This choice is not probably the bestfor hyperbolic NSWE, but for non-hydrostatic FNWD models it works very well due tothe inherent dispersive regularization property of solutions. Moreover, on the predictorstage we compute directly the quantities H, u and v (instead of computing the fluxes)since they are needed to compute the coefficients along with the right hand side F ofEq. (3.1).

Let us describe briefly the numerical algorithm we use to solve the extended system ofequations (2.1)-(2.3), (3.1) (for more details see also [34,46]). At the initial moment of timewe are given by the free surface elevation η(x, 0) and velocity vector u(x, 0). Moreover,if the bottom is not static, additionally we have to know also the quantities ht (x, 0) andhtt(x, 0). These data suffice to determine the initial distribution of the depth-integratedpressure℘ by solving numerically the elliptic equation (3.1). Finally, the dispersive pres-sure component on the bottom is computed by a finite difference analogue of Eq. (3.5).By recurrence, let us assume to we know the same data on the nth time layer t = tn : Hn ,un , ℘n and n . Then, we employ the predictor–corrector scheme, each time step of thisscheme consists of two stages. On the predictor stage we compute the quantities Hn+ 1

2 ,un+ 1

2 in cell centers as a solution of explicit discrete counterparts of Eqs. (2.6)-(2.8) (withright hand sides taken from the time layer tn). Then, one solves a difference equation

to determine ℘n+ 12 . The coefficients and the right hand side are evaluated using new

values Hn+ 12 and un+ 1

2 . From formula (3.5) one infers the value of n+ 12 . All the val-

ues computed at the predictor stage Hn+ 12 , un+ 1

2 , ℘n+ 12 and n+ 1

2 are then used at thecorrector stage to determine the new values Hn+1 and un+1 . At the corrector stage weemploy the conservative form of Eqs. (2.1)-(2.3). In the very last step we compute alsothe values of ℘n+1 and n+1 . The algorithm described becomes a spherical analogue ofthe well-known LAX–WENDROFF scheme if one neglects the dispersive terms.

An important property of the proposed numerical algorithm is its well-balanced char-acter if the bottom is steady (i.e. B ≡ 0) and the sphere is not too deformed (i.e. K > 0).In other words, it preserves exactly the so-called ‘lake-at-rest’ states where the fluid isat rest un ≡ 0 and the free surface is unperturbed ηn ≡ 0. Then, it can be rigorouslyshown that this particular state will be preserved in the following layer t = tn+1 as well.It is achieved by balanced discretizations of left and right hand sides in the momentumequations (2.2), (2.3). This task is not a priori trivial since the equilibrium free surfaceshape is not spherical due to Earth’s rotation effects [43].

4 Numerical illustrations

Currently, there is a well-established set of test problems [81] which are routinely used tovalidate numerical codes for tsunami propagation and run-up. These tests can be used


also for inter-comparison of various algorithms in 1D and 2D [38]. However, currently,there do not exist such (generally admitted) tests for nonlinear dispersive wave modelson a rotating sphere. The material presented below can be considered as a further effortto constitute such a database.

4.1 Wave propagation over a flat rotating sphere

Consider a simple bounded spherical domain which occupies the region from 100 to300 from the West to the East and from −60 to 65 from the South to the North. Fromnow on we use for simplicity the geographical latitude ϕ ≡ π

2 − θ instead of the variableθ . The considered domain is depicted in Fig. 2 and contains a large portion of the PACIFIC

OCEAN, excluding, of course, the poles (see restriction (2.4)). The idealization consists inthe fact that we assume the (undisturbed) water depth is constant, i.e. h ≡ 4 km. Theinitial condition consists of a GAUSSIAN-shaped bump put on the free surface

η(λ, ϕ, 0) = α0e−ρ2(λ, ϕ) , (4.1)

with zero velocity field in the fluid bulk. Function ρ(λ, ϕ) is a great-circle distance be-tween the points (λ, ϕ) and (λ0, ϕ0), i.e.

ρ(λ, ϕ)def:= R·arccos

cos ϕ cosϕ0 cos(λ − λ0) + sinϕ sinϕ0)

. (4.2)

In our numerical simulations we take the initial amplitude α0 = 5 m, the GAUSSIAN

center is located at(λ0, ϕ0

)=

(280,−40

). The parameter is chosen from three values

8×10−10 , 8×10−11 and 8×10−12m

−2. It corresponds to effective linear source sizes equalapproximatively to W1 ≈ 107.3km , W2 ≈ 339km and W3 ≈ 1073km respectively. Theeffective source size is defined as the diameter of the circle S10 serving as the level-set α0

10of the initial free surface elevation η(λ, ϕ, 0), i.e.

S10def:=

(λ, ϕ) |η(λ, ϕ, 0) =α0

10

.

On the boundary of the computational domain we prescribed SOMMERFELD-type non-radiation boundary conditions [34]. In our opinion, this initial condition has the advan-tage of being symmetric, comparing to the asymmetric source proposed in [51]. Indeed,if one neglects the EARTH rotation effect (i.e. Ω ≡ 0), our initial condition will gener-ate symmetric solutions in the form of concentric circles drawn on sphere’s surface. Ifone does not observe them numerically, it should be the first red flag. In the presence ofEARTH’s rotation (i.e. Ω > 0), the deviation of wave fronts from concentric circles fort > 0 characterizes CORIOLIS’s force effects (see Section 4.1.2).

Oscillations of the free surface were recorded in our simulations by two syntheticwave gauges located in points M1 =

(200, 0

)and M2 =

(120, 45

)(see Fig. 2). All

simulations were run with the resolution 6001×3751 of nodes (unless explicitly statedto the contrary). The physical simulation time was set to T = 30 h. In our numerical


(a) (b)

Figure 2: The omputational domain and free surfa e elevations omputed with the FNWD model for the sour e

W3 at t = 6h (a) and t = 23h (b). The lo ations of two syntheti wave gauges are depi ted with the symbol

⋆. The enter of the sour e region is shown with symbol .

experiments we observed that the CPU time for NSWE runs is about five times less thanFNWD computations. Sequential FNWD runs took about 6 days. This gives the first ideaof the ‘price’ we pay to have non-hydrostatic effects.

4.1.1 Sphericity effects

The effects of EARTH’s sphericity are studied by performing direct comparisons betweenour FNWD spherical model (2.1)-(2.3) and the same FNWD on the plane (see Part I [44]for the derivation and Part II [46] for the numerics). In the plane case the initial conditionwas constructed in order to have the same linear sizes W1,2,3 as in the spherical case.Namely, it is given by formula (4.1) with function ρ(x,y) replaced by the EUCLIDEAN

distance to the center(

x0,y0)

:

ρ(x,y) =√

(x − x0)2 + (y − y0)2 .

The computational domain was a plane rectangle with sides lengths approximativelyequal to those of the spherical rectangle depicted in Fig. 2. The source was located inthe point

(x0,y0

)such that the distance to the South–West corner is the same as in the

spherical configuration. Synthetic wave gauges M1,2 were located on the plane in orderto preserve the distance from the source. Moreover, in order to isolate sphericity effects,we turn off EARTH’s rotation in computations presented in this Section, i.e. Ω ≡ 0.

In Fig. 3 one can see the synthetic gauge records computed with the spherical and‘flat’ FNWD models. It can be clearly seen that sphericity effects become more and moreimportant when the source size W increases. We notice also that in all cases the waveamplitudes predicted with the spherical FNWD model are higher than in ‘flat’ FNWDcomputations. In general, the differences recorded at wave gauge M1 are rather moder-ate. However, when we look farther from the source, e.g. at location M2 the discrepanciesbecome flagrant — the difference in wave amplitudes may reach easily several times (as


(a) (b) (c)

(d) (e) (f)

Figure 3: Syntheti wave gauge re ords registered in points M1(a ) and M2(d f). FNWD predi tions on a

sphere are given with dashed lines (1) and the at model is depi ted with the solid line (2). The ee tive linear

sour e sizes are W1(a, d), W2(b, e) and W3( , f).

always the ‘flat’ model underestimates the wave). The explanation of this phenomenonis purely geometrical. In the plane case the geometrical spreading along the rays start-ing at the origin is monotonically decreasing (the wave amplitude decreases like O

(r−

12)

,where r is the distance from the source). On the other hand, in the spherical geometrythe rays starting from the origin

(λ0, ϕ0

)go along initially divergent great circles that in-

tersect in a diametrically opposite point to(λ0, ϕ0

)(this point has coordinates

(80, 40

)

in our case). As a result, the amplitude first decreases in geometrically divergent areas,but then there is a wave focusing phenomenon when the rays convergent in one point.A few such (divergent/convergent) rays are depicted in Fig. 2 with solid lines emanatingfrom the point

(λ0, ϕ0

). We intentionally put a wave gauge into the point M2 close to the

focusing area in order to illustrate this phenomenon. Otherwise, the amplification couldbe made even larger.

Such amplification phenomena should take place, in principle, on the whole sphere.It did not happen in our numerical simulation since we employ SOMMERFELD-type non-radiation boundary conditions. It creates sub-regions which are not attainable by the


waves. It happens since the rays emanating from the origin(λ0, ϕ0

)cross the boundary

of the computational domain (and, thus, the waves propagating along these rays arelost). To give an example of such ‘shaded’ regions we can mention a neighbourhood ofthe point

(100,−60

). That is why the wave field looses its symmetry as it can be seen

in Fig. 2(b): the edges of the wave front get smeared due to the escape of informationthrough the boundaries. On the other hand, in Fig. 2(a) we show the free surface profileat earlier times where boundary effects did not affect yet the wave front which conserveda circle-like shape.

We should mention an earlier work [13] where the importance of sphericity effectswas also underlined. In that work the authors employed a WNWD model to simulate theIndian Ocean tsunami of the 25th December 2004 [80]. The computational domain wassignificantly smaller than in our computations presented above (no more than 40 in eachdirection). That is why the sphericity effects were less pronounced than in Fig. 3. Moreprecisely, only 30% discrepancy was reported in [13] comparing to the ‘flat’ computation.

4.1.2 Coriolis effects

Let us estimate now the effect of EARTH’s rotation on the wave propagation process. Themain effect comes from the CORIOLIS force induced by the rotation of our planet. Thisforce appears in the right hand sides of Eqs. (2.2), (2.3) and (3.1). In Fig. 4 we showsynthetic wave gauge records (for precisely the same test case and recorded in the samelocations M1,2 as described in previous Section 4.1.1) with (Ω = 7.29×10−5

s−1) and

without (Ω = 0) EARTH’s rotation. We also considered initial conditions of differentspatial extents W1,2,3 . In particular, we can see that CORIOLIS’s force effect also increaseswith the source region size. However, EARTH’s rotation seems to reduce somehow waveamplitudes (i.e. elevation waves are decreased and depression waves are increased inabsolute value). We conclude also that CORIOLIS’s force can alter significantly only thewaves travelling at large distances.

Numerical simulations on a sphere, which rotates faster than EARTH, show that CORI-OLIS and centrifugal forces produce also much more visible effects on the wave propa-gation. In particular, important residual vortices remain in the generation region and thewave propagation speed is also reduced.

Concerning earlier investigations conducted in the framework of WNWD models, itwas reported in [13] that CORIOLIS’s force may change the maximal wave amplitude upto 15%, in [57] — 1.5 – 2.5% and in [51] — up to 5%. Our results generally agree withthese findings for corresponding source sizes. In the latter reference it is mentioned alsothat CORIOLIS’s force influence increases with source extent and it retains a portion of theinitial perturbation in the source region, thus contributing to the formation of the residualwave field [63].

4.1.3 Dispersive effects

In order to estimate the contribution of the frequency dispersion effects on wave prop-agation, we are going to compare numerical predictions obtained with the FNWD and


(a) (b) (c)

(d) (e) (f)

Figure 4: Syntheti wave gauge re ords registered in points M1(a ) and M2(d f). FNWD predi tions on a

sphere with Coriolis's for e are shown with dashed lines (1) and without Coriolis is depi ted with the solid

line (2). The ee tive linear sour e sizes are W1(a, d), W2(b, e) and W3( , f).

(hydrostatic non-dispersive) NSWE models on a rotating sphere§. Both codes are fedwith the same initial condition (4.2) as described above. In these simulations we use afine grid with the angular resolution of 40′′ in order to resolve numerically shorter wavecomponents. As it is shown in Fig. 5(a), FNWD model generates a dispersive tail be-hind the main wave front. The tail consists of shorter waves with smaller (decreasing)amplitudes. Obviously, the NSWE model does not reproduce this effect (see Fig. 5(b)).

Computations based on the FNWD model show that dispersive effects are more pro-nounced for more compact initial perturbations. Signals in wave gauges (see Fig. 4(a,d))show that the initial condition of effective width W1 generates a dispersive tail quasi-absent in W2 and inexistent in W3 . Another interesting particularity of dispersive wavepropagation is the fact that during long distances the maximal amplitude may move intothe dispersive tail. In other words, the amplitude of the first wave may becomes smallerthan amplitude of waves from the dispersive tail. Moreover, the number of the highest

§We use the same angular velocity Ω = 7.29×10−5s−1 .


(a) (b)

Figure 5: Zoom on the wave eld after t = 5h of free propagation over a sphere omputed with FNWD (a)

and NSWE (b) for the initial ondition W1 .

wave may increase with the propagation distance [68]. A similar effect was observed inthe framework of WNWD models in [28, 57, 58].

Moreover, the influence of dispersion is known to grow with the traveled distance[2, 72]. It can be also observed in Fig. 4. For example, for the initial condition of the sizeW2 , the frequency dispersion effect is not apparent yet on wave gauge M1 (see Fig. 4(b),line 1), but it starts to emerge in gauge M2 (see Fig. 4(e), line 1). Concerning the source ofsize W1 , here the dispersion becomes to play its rôle even before the point M1 . In order toestimate better the distance on which dispersive effects may become apparent, we createadditional synthetic wave gauges in points

Mi

6i=3 with coordinates

(λi, ϕ i

)6i=3 . The

longitude is taken to be that of the source center, i.e. λi ≡ λ0 = 280 , ∀i ∈ 3, .. . ,6, while thelatitude takes the following values: ϕ3 = −35 , ϕ4 = −30 , ϕ5 = −20 , ϕ6 = 0 . Thus,these wave gauges are located much closer to the source

(λ0, ϕ0

)than M1 . The recorded

data is represented in Fig. 6. In panels 6(a – c) one can see that for smallest source W1 thedispersion is already fully developed in gauge M5 (and becomes apparent in gauge M4),i.e. at the distance of ≈ 2200 km from the source. Indeed, in panel 6(c) the second waveof depression has the amplitude larger than the first wave. One can notice in generalthat the dispersion always yields a slight reduction of the leading wave (compare withNSWE curves in dash dotted lines (2)). To our knowledge this fact was first reported innumerical simulations in [10]. Concerning the initial condition of size W2 , the dispersiondoes not seem to appear even at the point M6 , located ≈ 4400 km from the source (seeFig. 6(d – f )). As a result, we conclude that, at least for tsunami applications, the frequencydispersion effect is mainly determined by the size of the generation region and thus, bythe wavelength of initially excited waves. The dispersion effect is decreasing when thesource extent increases. A common point between FNWD and NSWE models is that on


(a) (b) (c)

(d) (e) (f)

Figure 6: Syntheti wave gauge re ords predi ted by FNWD (1) and NSWE (2) spheri al models for initial

sour es of sizes W1(a ) and W2(d f). Wave gauges are lo ated in M3(a), M4(b, d), M5( , e) and M6(f).

relatively short distances the period of the main wave increases and the wave amplitudedecreases.

4.1.4 Some rationale on the dispersion

In order to estimate approximatively the distance ℓd on which dispersive effects becomesignificant for a given wavelength λ we consider a model situation. For simplicity weassume the bottom to be even with constant depth d. Let us assume also that wavedynamics is described by the so-called linearized BENJAMIN–BONA–MAHONY (BBM)equation [5, 69]:

ηt + υ0 ηx = νηxxt ,

which can be obtained from linearized FNWD plane model reported in [46]. Here

υ0def:=

√

gd is the linear long wave speed and νdef:= d2

6 . For this equation, the disper-sion relation ω(κ), the phase speeds ν and νκ , corresponding to the given wavelength λ


and wavenumber κ respectively, are given by the following formulas:

ω(κ) =υ0κ

1 + νκ2 , ν =υ0

1 + ν( 2π

λ

)2 , νκ =υ0

1 + ν( 2π

λκ

)2 ,

where λκ

def:= 2π

κ.

During the time t the generated wave travels the distance ℓd = ν·t, while a shorterwave of length λκ < λ will travel a shorter distance

ℓκ = νκ ·t = ℓdνκ

ν< ℓd .

The difference in traveled distances is a manifestation of the frequency dispersion. Now,we assume that the wavelength of a separated wave is related to λ as

λκ = δλ, δ ∈ (0,1),

and we assume that by time t the dispersion had enough time to act, i.e.

ℓκ = ℓd − (1 + δ)λ

2.

From this equality we can determine ℓd :

ℓd

(

1 −νκ

ν

)

=1 + δ

2λ,

and if we substitute the expressions of phase speeds ν and νκ we obtain:

ℓd

1 −

υ0

1 + ν( 2πλκ

)2

υ0

1 + ν( 2π

λ

)2

=1 + δ

2λ.

By using the fact that λκ = δλ we obtain the final expression for the dispersion distance:

ℓd =1

2(1 − δ)

[

λ +3

2π2 δ2 λ

3

d2

]

≈1

2(1 − δ)

[

λ + 0.152δ2 λ3

d2

]

. (4.3)

The dependence of the dispersive distance ℓd on the wavelength λ is depicted in Fig. 7.Let us mention that an analogue of formula (4.3) was obtained earlier in [60,68] based

on the (linearized) KORTEWEG–DE VRIES equation [7, 53].


0 50 100 150 2000

1000

2000

3000

λ

, km

, km

1 2 3 4 5

6

d

Figure 7: Dependen e of the dispersion distan e ℓd on the wavelength for parameter δ = 0.33 and for water

depth d = 1km (1), d = 2km (2), d = 3km (3), d = 4km (4), d = 5km (5) and d = 6km (6).

Numerical application. In order to apply formula (4.3) (and check its validity) to ourresults, we have to estimate first the generated wavelength λ and parameter δ. We sawabove that for the initial condition of extent W1 the wave gauge M5 registered a well-separated secondary wave with the period about three times smaller. Thus, we can setδ = 0.33. Concerning the wavelength of the generated wave, in such estimations it isroughly identified¶ with the size of the generated region W1 [51]. Under these assump-tions, Eq. (4.3) gives us the dispersive distance ℓd = 1058 km. Thus, we can concludethat dispersive effects should be apparent in wave gauge M5 , which is in the perfectagreement with our numerical simulations. Let us take a source region with a larger hor-izontal extent W2 . If we take δ = 0.33 (as above) and λ = 339 km, formula (4.3) givesus ℓd = 30139km . Thus, we conclude that dispersive effects will not be seen even in thefarthest wave gauge M2 .

Alternative approaches. There exist other criteria of the importance of frequency dis-persion effects. One popular criterium is the so-called KAJIURA number Ka employed,for example, in [51]:

Kadef:=

( 6d

ℓ

) 13·W

d,

where W is the source region width and ℓ is the distance traveled by the wave. It isbelieved that dispersive effects manifest if Ka < 4. Let us check it against our numerical

¶However, in the general case this approximation might be rather poor [68].


simulations. In the case depicted in Fig. 6(c) we have W = 107.3 km and ℓ = 2200 km .Thus, Ka ≈ 6 > 4. According, to KAJIURA’s criterium, the dispersion should not appearyet at the point M5 . However, it contradicts clearly our direct numerical simulationsshown in Fig. 6(c). It was already noticed in [28] that criterium Ka < 4 is too stringent.It was proposed instead to use a more sophisticated criterium based on the normalizeddispersion time:

ϑdef:= 6

√

gdd2

λ3 t.

If ϑ > 0.1 the wave propagation is considered to be dispersive. In our simulations t =12000 s. It is the time needed for the wave to travel to the location M5 . Thus, in ourcase ϑ ≈ 0.18 > 0.1 and the dispersion should be already visible in location M5 for thesource size W1 , which is in perfect agreement with our synthetic wave record (see againFig. 6(c)).

In our opinion, the use of the criterium (4.3) based on the dispersion distance ℓd ispreferable, since it gives directly an estimation of the distance, where the dispersive ef-fects will become non-negligible. Outside the circle of radius ℓd and centered at the sourcethe use of hydrostatic non-dispersive models is not advised.

Let us mention also a few earlier works where the importance of dispersive effectswas studied for real-world events using WNWD models. In [32, 39] the SUMATRA 2004INDIAN ocean tsunami was studied and it was shown that in the deep West part of theINDIAN ocean the discrepancy in wave amplitudes reaches 20% (between WNWD andNSWE models). For the same tsunami event the discrepancy of 60% was reported in [13].Numerical simulations of the TOHOKU 2011 tsunami event reported in [51] confirmedagain the difference of 60% between hydrostatic and non-hydrostatic model predictions.

4.2 Bulgarian 2007 tsunami

The basin of the BLACK SEA is subject to a relatively important seismic activity and thereexists a potential hazard of tsunami wave generation, which may be caused not only byearthquakes, but also by underwater landslides, which can be triggered even by weakseismic events. Geophysical surveys show that large portions of the BLACK SEA conti-nental shelf contain unstable masses [23], which have to be taken into account while as-sessing tsunami hazard for the population or and underwater infrastructure. Moreover,some past events are well documented [41].

Some anomalous oscillations of the sea level were registered on the 7th of May 2007at Bulgarian coasts. In [70] it was conjectured that a landslide may have provoked theseoscillations and the authors considered four possible locations of this hypothetical land-slide. In all these cases the landslide started at the water depth about 100m and at thedistances of 30 – 50km from the coast. The suggested volume of the landslide is between30 and 60 millions of m3 . Landslide thickness is about 20 – 40m. In [70] the authors rep-resented the landslide as a system of interconnected solid blocks which can move alongthe slope under the force of gravity whose action is compensated by frictional effects.


The hydrodynamics was described by NSWE incorporating some viscous effects solvedwith the Finite Element Method (FEM). For all four considered cases simulations showthat the landslide achieves the speed of about 20m/s in 200 – 300s after the beginning ofthe motion. Landslide’s motion stops at the depth around 1000m after running out about20km from its initial position.

The hypothesis of landslide mechanism is studied by confronting numerical predic-tions with eyewitness reports and coastal wave gauge data. In particular, we know thevalues of lowest (i.e. negative) and highest amplitudes for seven locations along the coastrespectively:

Shabla −1.5m, 0.9m;

Bolata −1.3m, 0.9m;

Dalboka −2.0m, 1.2m;

Kavarna −1.8m, 0.9m;

Balchik −1.5m, 1.2m;

Varna −0.7m, 0.4m;

Galata −0.2m, 0.1m.

The comparisons from [70] show that one can seemingly adopt the landslide mechanismhypothesis. However, even the most plausible landslide scenario (among four consid-ered) does not give a satisfactory agreement with all available field data. Moreover, themaximal synthetic amplitudes are shifted to the South (towards EMINE), which was notobserved during the real event.

In a companion study [87] the authors investigated also the hypothesis of a meteo-tsunami responsible of anomalous waves recorded on the 7th of May 2007 at Bulgariancoasts. Their analysis showed that the weather conditions could provoke anomalouswaves near BULGARIAN coasts. The numerical simulations show again a good agreementof maximal amplitudes at some locations, even if we cannot speak yet about a goodgeneral agreement. In particular, numerical predictions seriously underestimate waveamplitudes in Northern parts of the coast such as SHABLA and overestimate them inSouthern ones (e.g. large waves were not observed in BURGAS Bay, but they existed innumerical predictions).

In the present study we continue to develop the landslide-generated hypothesis ofanomalous waves. In contrast to the previous study [70], we employ the FNWD modelpresented above and the landslide will be modeled using the quasi-deformable bodyparadigm [4]. The driving force was taken as the sum of the gravity, buoyancy, fric-tion and water drag forces acting on elementary volumes. Quasi-deformability propertymeans that the landslide can deform in order to follow complex bathymetry profiles by


preserving the general shape (see [4, 20, 21] for more details). However, the deforma-tion process is such that the horizontal components of the velocity vector are the samethroughout the sliding body (as in the absolutely rigid case). This model has been vali-dated against experimental data [34] and direct numerical simulations of the free surfacehydrodynamics [36]. Moreover, it was already successfully applied to study numeri-cally a real world tsunami which occurred in PAPUA NEW GUINEA on the 17th of July1998 [48]. Important parameters, which enter in our landslide model and, thus, that haveto be prescribed are:

V landslide volume;

Cw added mass coefficient;

Cd drag coefficient;

C f ≡ tanθ∗ friction coefficient and θ∗ is the friction angle;

γdef:= ρs

ρw> 1 ratio between water ρw and sliding mass ρs densities.

The initial shape of the landslide is given by the following formula:

h0s (x,y) =

T

4

[

1+ cos( 2π(x − x0

c)

Bx

)]

·

[

1+ cos( 2π(y− y0

c )

By

)]

, (x,y) ∈ D0 ,

0, (x,y) /∈ D0 ,

where D0 =[

x0c − Bx

2 , x0c + Bx

2

]×[y0

c −By

2 ,y0c +

By

2

]is the domain occupied by the

sliding mass, Bx,y are horizontal extensions of the landslide along the axes Ox and Oyrespectively,

(x0

c ,y0c

)is the position of its barycenter and T is its thickness.

Remark 4.1. For the sake of notation compactness, in the landslide description above weused CARTESIAN coordinates. This approximation is valid since landslide size is smallenough to ‘feel’ EARTH’s sphericity. We place the origin in the left side center of thespherical rectangular computational domain, i.e. in the point

(27, 43

). Then, the local

CARTESIAN coordinates are introduced in the following way:

x = Rπ

180(λ − 27)cos

( π

18043

)

, y = Rπ

180(ϕ − 43).

If(λ

0c , ϕ0

c

)are spherical coordinates of the landslide barycenter, then its local CARTESIAN

coordinates(

x0c ,y0

c

)are computed accordingly:

x0c = R

π

180(λ0

c − 27)cos( π

18043

)

, y0c = R

π

180(ϕ0

c − 43).


Table 1: Physi al parameters used to simulate the hypotheti al landslide motion during Bulgarian tsunami of

the 7

th

of May 2007.

Parameter Value

Added mass coefficient, Cw 1.0Drag coefficient, Cd 1.0Densities ratio, γ 2.0Friction angle, θ∗ 1

Landslide thickness, T 40mLandslide length, Bx 2500mLandslide width, By 2500mLandslide volume, V 62.5×106

m3

During the modelling of landslide events in the BLACK SEA we used the parametersof some historical events [41]. We noticed that the most sensitive parameter is the initiallocation of the landslide. That is why in the present study we focus specifically on thisaspect in order to shed some light on this unknown parameter. In this perspective wechose 40 different initial locations along the BULGARIAN coastline which were locatedmainly at the depth of 200m, 1000m and 1500m. These locations are depicted with blackrectangles in Fig. 8. Other parameters are given in Table 1. The volume V of the landslidein our simulations is equal to 62.5×106

m3 , which is close to the value used in [70]. We

notice that there are two competing effects in our problem. If we increase the initiallandslide depth, the amplitude of generated waves will be seriously reduced. However,this effect can be compensated by increasing the landslide thickness T . In general, theamplitude of waves is proportional to T .

In these numerical simulations we use the finest angular resolution (in this section)of 3.75′′ , since the domain is relatively compact. It corresponds to the grid of 2881×1921nodes. The bathymetry data was obtained by applying bilinear interpolation to dataretrieved from “GEBCO One Minute Grid – 2008”. The computational (CPU) time of eachrun was about 35h for this resolution. Some information on the wave propagation can beobtained using the so-called radiation diagrams, which represents the spatial distributionof maximal and minimal wave amplitudes‖ during the whole simulation time. Aftercomputing the first 40 scenarii (marked with little black rectangles in Fig. 8), we coulddelimit∗∗ the area where the hypothetical landslide could take place. Then, in the secondtime, this area was refined with additional 171 initial landslide locations marked withpluses in Fig. 8. In this way, by comparing simulation results with available field data,we could choose the most probable scenario of the initial location of the landslide. Finally,we performed the third optimisation cycle in order to determine the most likely landslide

‖Here we mean maximal positive and minimal negative waves with respect to the still water level.∗∗Here we stress again that the computational domain was the same, but we delimit the search area for thelandslide initial position.


Figure 8: Computational domain and the distribution of minimal amplitudes of the waves generated by the

optimal landslide (4.4) omputed during the rst 3 h of the physi al propagation time. With bla k lines we

represent iso-values of the bathymetry. With symbols ⋆ we denote oastal towns where we know the maximal

and minimal wave amplitudes. Little bla k re tangles and pluses denote various initial positions of the landslide

onsidered in our study. The green line shows the most probably landslide traje tory. Finally, the little bla k

arrow shows the starting point and dire tion along the oastline where we re ord maximal and minimal wave

amplitudes.

thickness T . The optimal parameters are given here:

λ0c = 28.7341 , ϕ0

c = 42.8871 , T = 320m, Bx = By = 2500m. (4.4)

The radiation diagram for this tsunami event is shown in Fig. 8. One can see, in partic-ular, that the radiation of minimal (negative) amplitudes is directed towards the coastaltowns where the most significant oscillations of the sea level were observed. We noticealso that in our simulations we do not obtain a two-tongue structure predicted in [70, Fig-ure 2]. However, such radiation diagrams are very sensitive to the initial location of thelandslide. For some starting points we observed (as in an earlier work [48]), for exam-ple, an abrupt termination of landslide motion, which affected quite a lot the radiationdiagram.

Minimal and maximal wave amplitudes recorded along the BULGARIAN coast areshown in Fig. 9. The starting point of the path along the coastline is shown with a littleblack arrow in Fig. 8. We mention also that the distance is computed on our grid (thus, thecoastline is approximated in fine by a polygon). Hence, there might be little discrepancies(in the sense of overestimations) with ‘real-world’ distances. Fig. 9 shows an overall goodagreement with field data. Moreover, in contrast to the previous study [70], our scenario


Figure 9: Minimal and maximal os illations of the free surfa e along the Bulgarian oastline re orded during

the whole numeri al simulation of the most likely landslide event. Blue solid lines are numeri al predi tions and

bla k ir les represent eld data.

does not trigger large wave amplitudes in Southern parts of the BULGARIAN coast.We have to mention that a few other scenarii gave comparable agreement with field

data. However, if we look carefully at corresponding landslide trajectories, they all passthrough the termination point of the optimal landslide†† (4.4). Consequently, we can onlysuggest to study this area of the BLACK SEA in the perspective to discover eventually thedeposits of this past landslide event.

We underline also that the geometrical extensions of the ‘optimal’ landslide (4.4) canbe changed without loosing too much the good agreement with observations demon-strated in Fig. 9. For example, we used the following parameters to trigger a tsunamiwave with similar amplitudes in observation points:

λ0c = 28.7341 , ϕ0

c = 42.8871 , T = 110m, Bx = By = 5000m. (4.5)

In general, such modifications may alter seriously the landslide trajectory and velocity.However, in this particular case the new trajectory followed closely the green line de-picted in Fig. 8.


The influence of the frequency dispersion in this particular tsunami event will be esti-mated by computing the absolute and relative‡‡ differences between radiation diagrams

††We have to say that even the optimal landslide may move farther if we slightly decrease the friction angleθ⋆ .‡‡In the relative difference we divide by the magnitude of the NSWE prediction.


(a) (b)

Figure 10: The spatial distribution of absolute (a) and relative (b) dieren es in the maximal positive wave

amplitudes omputed a ording to FNWD and NSWE models for the optimal landslide (4.4).

(i.e. maximal amplitudes) computed with FNWD and NSWE models. In this way, inFig. 10 we show how the incorporation of non-hydrostatic effects modifies extreme (pos-itive) wave amplitudes. In particular, one can see that relative differences can reach upto 70% in deep parts of the BLACK SEA. We computed differences of radiative diagramsfor the nearly-optimal landslide (4.5) (with smaller thickness T) and it seems that the dis-persion plays smaller effect in that case. In general, when there is an abrupt terminationof landslide motion, the differences between dispersive and non-dispersive predictionsincrease. Our computations show also that the period of principal wave components isbetween 250 – 400s and waves of maximal (absolute) amplitude do not always come first(in a good agreement with observations). To save the space we do not provide these nu-merical results here. The main goal of this Section is to demonstrate that FNWD modelscan be successfully used to study real world events on all scales from regional to globalones. Even better, today one can successfully perform parametric studies with sphericalFNWD models: in order to determine the optimal landslide parameters given in (4.4), wehad to run 211 scenarii in total.

4.3 Chilean 2010 tsunami

In order to illustrate the application of our spherical FNWD model to a real-world largescale seismically generated tsunami event we consider the CHILEAN tsunami which tookplace on the 27th of February 2010. Earthquake epicenter was located under the Ocean117 km to the North from CONCEPCIÓN at the depth of about 35 km below the bottom.This event was estimated to have the seismic moment magnitude Mw = 8.8. This earth-quake generated a tsunami wave, which was observed in the whole PACIFIC OCEAN. Themost important aspect for us is that this wave was registered at DART buoys. Many sci-entific works are devoted to the investigation of this particular event. Here we mentiona few numerical studies which go along the lines of our own work [1,67,89]. Contrary to


Figure 11: The omputational domain (a) and the initial wave elevation (b) omputed a ording to USGS

inversion of the Chilean 2010 earthquake. Symbol ⋆ on the right panel shows the earthquake epi enter.

the catastrophic TOHOKU 2011 event, where researchers had to introduce local landslidehypothesis in order to explain some extreme run-up values [83], CHILEAN 2010 eventseems to be purely seismic since the available data on this tsunami can be reproducedfairly well starting from the initial water column disturbances caused by the earthquakesolely.

In order to reconstruct the displacement field of the EARTH surface, some authors useGPS data [14,61,86]. Later, these seismic scenarii were tested in [1] to confront them withavailable tsunami field data. The final agreement quality dependent on the chosen sce-nario. In the present work we consider the fourth alternative proposed by USGS. Earthsurface displacements were reconstructed using the celebrated OKADA solution [65, 66].Then, this displacement was transferred to the free surface as the initial condition for ourhydrodynamic computations. This tsunami generation procedure is known as the ‘pas-sive generation’ approach [19, 42]. We would like to mention that there are noticeablediscrepancies among all these seismic inversions. Thus, there is an uncertainty in the ini-tial condition for tsunami wave propagation [15–17]. Our choice for the USGS inversioncan be explained essentially by the immediate availability of their data through their website.

The computational domain used in our simulations covers a significant portion of thePACIFIC ocean —

[199, 300

]×[−60, 5

]. We used a 1′ grid and the bathymetry data

was taken from “The GEBCO One Minute Grid — 2008”. The use of finer grids or com-putation in significantly larger domains does not seem to be feasible with serial codes(see Section 5.2). In order to validate our code we present the comparisons of predictedtsunami waves against three DART buoys: DART–32411, DART–32412 and DART–51406.The locations of these buoys are shown in Fig. 11(a) and the initial condition is repre-sented in Fig. 11(b). Comparisons of FNWD predictions against aforementioned DARTdata is shown in Fig. 12. Buoys data was downloaded from the National Oceanic and


Figure 12: Comparison of our numeri al predi tions with the spheri al FNWD model with DART data.

Atmospheric Administration (NOAA) web site. In the case of DART–32411 and DART–51406 buoys a vertical translation of data was needed to adjust the still water level. Onecan see an overall good agreement in Fig. 12 between our simulation and real-world data.The first oscillations present in DART data for t < 1h do not seem to be related to thestudied tsunami event, since the wave did not have enough time to travel from the sourceregion to the observation point. One can see also that synthetic records have somehowsmaller amplitudes. It can be related to our choice of the initial condition (USGS) whichdid not take into account tsunami-related constraints during the inversion process [1].


In order to estimate the influence of dispersive effects in this particular tsunami event, weperform two simulations — with FNWD and NSWE models. Moreover, we consider onecase with the real bathymetry data and another one with an even bottom of constant d =4km depth. Radiation diagrams for all these four cases are shown in Fig. 13. One can seethat FNWD and NSWE predict significantly different radiation diagrams∗. The differencebecomes even more flagrant in the idealized constant depth case†. Bottom irregularities

∗Compare the top panels 13(a, b), which show FNWD result with lower panels 13(c, d) representing NSWEpredictions.†Compare panels 13(b) and 13(d).


Figure 13: The distribution of maximal positive wave amplitudes predi ted with FNWD model (a, b) and

NSWE ( , d). The real bathymetry data is used in omputations (a, ), while the even bottom of onstant

depth d = 4km is used in (b, d).

contribute equally to radiation diagrams even if they fail to alter the maximal radiationdirection (at least in this particular tsunami event). It seems that here the initial conditionhas the dominant rôle in shaping the triggered tsunami wave.

In order to highlight the differences between NSWE and FNWD models predictions,we present in Fig. 14 the absolute and relative‡ differences among the correspondingradiation diagrams. The biggest absolute differences are concentrated along the mainradiation direction and NSWE model seems to overestimate substantially the wave am-plitude. The picture of relative differences has a much more complex structure even inthe idealized case. The largest relative differences attain easily 60% not only along themain radiation direction, but also to the South from the epicenter. We can only concludethat the frequency dispersion has to be taken into account in this event. However, thedispersion effect may vary with the tsunami initial condition [35]. Consequently, it is notexcluded that for other seismic inversions the initial free surface shape may change suchthat the dispersion will play a more modest rôle.

‡While computing the relative difference, we divide by the magnitude of the NSWE prediction as we did itabove in Section 4.2.1.


Figure 14: The absolute (a, b) and relative ( , d) dieren es among radiation diagrams omputed a ording to

FNWD and NSWE models on a real bathymetry (a, ) and on an even onstant bottom (b, d).

5 Discussion

After the numerical developments and illustrations presented above, we finish thismanuscript by outlining the main conclusions and perspectives of the present study andthis series of papers in general.

5.1 Conclusions

In this work we presented a numerical algorithm to simulate the generation and prop-agation of long surface waves in the framework of a fully nonlinear weakly dispersivemodel on a globally spherical domain. Our model includes the rotation effects (CORIOLIS

and centrifugal forces) of the EARTH. Using a judicious choice of the still water level, wecould ‘hide’ the terms corresponding to the centrifugal force. However, if the still waterlevel is assumed to be spherical, then these terms have to appear in governing equations.

Our numerical method is based on the numerical solution of the extended system,which consists of an elliptic equation to determine the dispersive component ℘ of thedepth-integrated pressure and of a quasi-linear first order hyperbolic part common withNonlinear Shallow Water Equations (NSWE). Both parts are coupled via source terms inthe right hand of hyperbolic equations. These source terms come from non-hydrostatic


effects. Thus, one can use the favourite numerical method for elliptic and hyperbolicproblems. For the elliptic part we employed the finite differences, while hyperbolic equa-tions were discretized with a two-step predictor–corrector scheme. On every stage of thisscheme we solve both sub-problems.

The performance of the proposed algorithm is illustrated on several test cases. First,we consider an idealized situation of wave propagation over an even bottom. However,in this ideal setting we study the importance of sphericity, rotation, CORIOLIS and disper-sion effects. For instance, we showed that the force of CORIOLIS becomes important onlyon large propagation distances (unless the angular velocity Ω is increased) and the rota-tion reduces somehow the maximal positive wave amplitudes (but only positive). Thefrequency dispersion also appears at rather large distances, but it depends greatly on thesize of the initial free surface elevation, i.e. more compact sources generate more dis-persive waves. Contrary to the dispersion, the CORIOLIS effect becomes more importantwhen we increase the generation area.

The question of dispersive effects importance is even more complex, since it can varywith the source size, but also with the source shape (i.e. the initial condition). Conse-quently, for real-world problems a special investigation is needed in each particular case.It does not seem realistic that using simple criteria today we can select the most perti-nent hydrodynamic model. In practical problems the bathymetry profile may have animportant effect as well.

In the present study we did not rise the question of the numerical dispersion at all.Here we can mention that for grid resolutions of 40′′ and smaller, the numerical dis-persion seems to be completely negligible comparing to the physical one (for the initialcondition of the horizontal extension W1 , for shorter waves the grid spacing has to befurther reduced, of course). However, we are not sure that a weakly dispersive modelis able to reproduce such waves with acceptable accuracy, without even speaking of theinherent computational complexity on such fine grids.

5.1.1 General conclusions

In this series of papers our main goal was to present a unified framework to modellingand numerical simulations of nonlinear dispersive waves. Indeed, in Part I [44] we pre-sented a generalized derivation of dispersive wave models on a plane and the base modelcontained a free modeling variable — the dispersive component of the horizontal velocityvector. By making special choices of this free variable, we could recover some known andsome new models. Then, in Part III [43] the same approach was presented for globallyspherical geometries with a similar degree of freedom at our disposal. The derivationson a sphere are more technical but in Part III we really follow the main lines of Part I.Finally, there is a similar interplay between numerical Parts II [46] and IV [45] on theglobally flat and globally spherical geometries respectively. The key ingredient in bothcases consists in deriving an elliptic equation to determine the dispersive component ℘of the depth-integrated pressure. Then, the governing equations are decoupled into anelliptic and hyperbolic parts. For each part we apply the most suitable numerical method.


Even if the details are quite different in flat and spherical cases, the philosophy remainsinvariant. As we hope, the numerical tests presented in Parts with even numbers, areconvincing enough to show the operational qualities of the proposed algorithm.

5.2 Perspectives

One of the drawbacks of non-hydrostatic FNWD models is the inherent computationalcomplexity. At every time step we have to solve an elliptic equation at least once. Most ofruns presented in this study took about a week of CPU time to be completed in serial im-plementations. Consequently, in the future, we plan to develop a parallel version of theNLDSW_SPHERE code in order to be able to handle much faster even higher grid reso-lutions. After this step we could try to implement fully nonlinear models with improvedlinear dispersion relation properties derived in [43]. Another improvement would consistin a better representation of the shoreline. In the current implementation the shoreline isapproximated by a polygon with sides parallel to (spherical) coordinate axes. Perhaps,a local mesh refinement could improve this point but it would create other numericaldifficulties. Finally, the main point which remains to be solved is the development of agenuine run-up algorithm in the spirit of local analytical in time solutions [50], whichtakes into account the non-hydrostatic nature of FNWD governing equations.

Acknowledgments

This research was supported by RSCF project No 14–17–00219.We would like to thank also Professor Emmanuel AUDUSSE (Université Paris 13,

France) for rising the important question of boundary conditions in the SERRE–GREEN–NAGHDI equations. We did our best to explain our approach to this problem in the spher-ical case, which is described in Section 3.1.1.

A Derivation of the equation for non-hydrostatic pressure

component

In this Appendix we provide the complete derivation of the elliptic equation (3.1) forthe dispersive component ℘ of the depth-integrated pressure p. It is convenient to startwith the base model (2.1)-(2.3) written in the following equivalent form (see [43] for moredetails):

(JH)t +[JHu1]

λ+

[JHu2]

θ= 0, (A.1)

(JHv)t +[JHvu1]

λ+

[JHvu2]

θ+ g∇

(

JH2

2

)

= gH2

2∇J +

[

gH∇h + HS + ∇℘ − ∇h]

J, (A.2)


where v =(

v1,v2)

is the covariant velocity vector:

v1 = (Ω + u1)R2 sin2θ , v2 = R2u2 . (A.3)

Finally, the vector S =(0, s2

)with

s2 =[2Ωu1 + (u1)2]R2 sinθ cosθ .

Eq. (A.2) can be rewritten in a non-conservative form:

Dv = −g∇η +℘ − ∇h

H+ S, (A.4)

where Dv ≡ vt + u·∇v.As in the globally flat case, we shall express first the dispersive pressure component

on the bottom as a function of ℘ and other variables. From definitions (2.5) we have

=3℘2H

+H

4R2 , (A.5)

where the term R2 can be fully expanded:

R2def:= D2h ≡ D(Dh) = (Dh)t + u·∇(Dh)

=[ht + u·∇h

]

t+ u·∇

[ht + u·∇h

]

= htt + 2u·∇ht︸︷︷︸

def=: B

+ ut ·∇h + u·∇(u·∇h).

The term B contains all the terms involving the bottom motion. The last term in R2 canbe further transformed§ equivalently as

u·∇(u·∇h) =[(u·∇)u

]·∇h + u·

[(u·∇)∇h

].

Consequently,

R2 = B + ut ·∇h +[(u·∇)u

]·∇h + u·

[(u·∇)∇h

]

≡ B + (Du)·∇h + u·[(u·∇)∇h

], (A.6)

§Indeed,

u·∇(u·∇h) = u1[u1 hλ + u2 hθ

]

λ+ u2[u1 hλ + u2 hθ

]

θ

= u1 u1λ

hλ + (u1)2 hλλ + u1 u2λ

hθ + u1 u2 hλθ + u2 u1θ hλ + u2 u1 hλθ + u2 u2

θ hθ + (u2)2 hθθ

=(u1 u1

λ+ u2 u1

θ

)hλ +

(u1 u2

λ+ u2 u2

θ

)hθ + u1(u1(hλ)λ + u2(hλ)θ

)+ u2(u1(hθ)λ + u2(hθ)θ

)

=[(u·∇)u

]·∇h + u·

[(u·∇)∇h

].


where in component-wise form we have

Du1 = u1t + u1u1

λ + u2u1θ ,

Du2 = u2t + u1u2

λ + u2u2θ ,

(Du)·∇h =(u1

t + u1u1λ + u2u1

θ

)hλ +

(u2

t + u1u2λ + u2u2

θ

)hθ ,

u·[(u·∇)∇h

]= (u1)2hλλ + 2u1u2hλθ + (u2)2hθθ .

By substituting the last expression in (A.6) into Eq. (A.5) we obtain

=3℘2H

+H

4

B + (Du)·∇h + u·[(u·∇)∇h

]

. (A.7)

Now let us express the convective term Du in terms of Dv. According to formulas (A.3),the covariant vector of the velocity v can be rewritten as

v = ΩG + G ·u,

where

Gdef:=

(g11

0

)

, Gdef:=

(g11 00 g22

)

,

with g11 , g22 being covariant components of the metric tensor on a sphere [43], i.e.

g11 = R2 sin2 θ , g22 = R2 .

Then, we have

Dv = ΩDG + D(G·u) = u2ΩGθ + G·Du + u1u2

Gθ = G·Du + u2(Ω + u1)

Gθ , (A.8)

where

Gθ =

(2R2 sinθ cosθ

0

)

.

By inverting Eq. (A.8), we obtain:

Du = G−1 ·Dv − u2(

Ω + u1)G−1 ·Gθ ,

and using the non-conservative equation (A.4) we arrive to the required representation:

Du = G−1 ·

−g∇η +℘ − ∇h

H

+ G−1 ·Λ, (A.9)

with

G−1 =

(g11 00 g22

)

, g11 ≡1

g11=

1R2 sin2 θ

, g22 ≡1

g22=

1R2 ,

Λ =

(Λ1Λ2

)

= S − u2(Ω + u1)

Gθ = R2 sinθ cosθ

(−2u2

(Ω + u1

)

u1(

2Ω + u1)

)

.


Finally, by substituting expression (A.9) into Eq. (A.7) we obtain:

=3℘2H

+H

4

∇℘·∇h − |∇h|2

H+ Q

,

where

∇℘·∇h ≡ g11℘λhλ + g22℘

θ hθ =1

R2

[℘λhλ

sin2θ+ ℘

θ hθ

]

,

|∇h|2 ≡ g11h2λ + g22 h2

θ =1

R2

[h2λ

sin2θ+ h2

θ

]

,

Qdef:=

[Λ − g∇η

]·∇h + B + u·

((u·∇)∇h

),

∇η ·∇h ≡ g11 ηλhλ + g22 ηθ hθ =1

R2

[ηλhλ

sin2θ+ ηθ hθ

]

,

Λ·∇h ≡ g11Λ1hλ + g22

Λ2hθ

= cotθ[

−2u2(Ω + u1)hλ + u1(2Ω + u1)hθ sin2 θ

]

.

Consequently, the dispersive part of the fluid pressure at the bottom can be expressed interms of other variables as

=1Υ

6℘H

+ HQ + ∇℘·∇h

, (A.10)

where we introduced a new dependent variable

Υdef:= 4 + |∇h|2 .

Now we can proceed to the derivation of an equation for ℘. From definitions (2.5) itfollows that

℘ =H3

12R1 +

H

2, (A.11)

where R1 was defined as

R1def:= D(∇·u) − (∇·u)2 .

Let us transform the last expression for R1 using the definition of the divergence ∇·(·)


and material derivative D operators

D(∇·u) = (∇·u)t + u1(∇·u)λ + u2(∇·u)θ

= ∇·ut + u1

u1λ+

(Ju2

)

θ

J

λ

+ u2

u1λ+

(Ju2

)

θ

J

θ

= ∇·ut +(u1u1

λ

)

λ− (u1

λ)2 +

(u2u1

θ

)

λ− u2

λu1θ + u1

(Ju2

)

λθ

J+ u2

[(Ju2

)

θ

J

]

θ

= (u1t )λ

︸︷︷︸

⋆⋆ 1

+

(Ju2

t

)

θ

J+

[u1u1

λ + u2u1θ

]

λ︸︷︷︸

⋆⋆ 1

+

[u1

(Ju2

)

λ

]

θ

J−

u1θ

(Ju2

)

λ

J

+

[

u2

(Ju2

)

θ

J

]

θ

− u2θ

(Ju2

)

θ

J−

(u1λ

)2− u2

λu1θ .

It is not difficult to see that terms marked with (⋆⋆ 1) can be aggregated into (Du1)λ .Consequently, we have

D(∇·u) = (Du1)λ +

(Ju2

t

)

θ+

(Ju1u2

λ

)

θ

J︸︷︷︸

⋆⋆ 2

− u1θ u2

λ +

(Ju2u2

θ

)

θ

J︸︷︷︸

⋆⋆ 2

+

[Jθ (u

2)2]

θ

J−

[u2

(Ju2

)

θ

J2 Jθ +u2

θ

(Ju2

)

θ

J+ (u1

λ)2]

︸︷︷︸

⋆

− u2λu1

θ .

The terms marked above with (⋆⋆ 2) give another interesting combination:[J(u2

t + u1u2λ+ u2u2

θ

)]

θ

J≡

(JDu2

)

θ

J.

According to the definition of the divergence ∇·u, the term (⋆) can be transformed as

(u1λ)2 +

[ (Ju2

)

θ

J

]2

≡(∇·u

)2− 2u1

λ

(Ju2

)

θ

J.

Consequently, we can rewrite D(∇·u) using mentioned above simplifications:

D(∇·u) =(Du1)

λ+

(JDu2

)

θ

J− (∇·u)2

+ 2u1λu2

θ + 2u1λu2 Jθ

J+ 2u2u2

θ

Jθ

J+ (u2)2 Jθθ

J− 2u1

θ u2λ .

Finally, taking into account the fact that Jθθ ≡ −J, we obtain:

D(∇·u) = ∇·(Du) − (∇·u)2 + 2(u1λu2

θ − u1θ u2

λ

)+ 2u2(u1

λ + u2θ

)cotθ − (u2)2 .


By assembling all the ingredients together, we obtain the following expression for R1 :

R1 = ∇·(Du) − 2(∇·u)2 + 2(u1λu2

θ − u1θ u2

λ

)+ 2u2(u1

λ + u2θ

)cotθ − (u2)2 .

By substituting the last expression for R1 into Eq. (A.11) and using formulas (A.9), (A.10)for Du and correspondingly, we obtain the first version of the required equation for ℘:

℘ =H3

12

[

∇·

G−1 ·

(

Λ − g∇η +∇℘

H−

6∇h

H2Υ

℘ − ∇hQ

Υ−

(∇℘·∇h

)∇h

HΥ

)

− 2(∇·u)2 + 2(u1λ

u2θ − u1

θ u2λ

)+ 2u2(u1

λ+ u2

θ

)cotθ − (u2)2

]

+H

2Υ

6℘

H+ HQ + ∇℘·∇h

. (A.12)

We can notice that the multiplication of covariant vectors Λ, ∇η , ∇℘ and ∇h by matrixG−1 transforms them into contravariant vectors which enter into the definition of thedivergence operator, for example

∇·∇ηdef:=

(Jg11ηλ

)

λ+

(Jg22 ηθ

)

θ

J=

1R2 sinθ

[ ηλλ

sinθ+

(ηθ sinθ

)

θ

]

≡ ∆η ,

∇·Λdef:=

(Jg11

Λ1)

λ+

(Jg22

Λ2)

θ

J=

1R2 sinθ

[Λ1,λ

sinθ+

(Λ2 sinθ

)

θ

]

.

We can notice also that the following identity holds:

∇·

[6∇h

H2Υ℘]

=6

H2 Υ∇℘·∇h + 6℘∇·

[∇h

H2 Υ

]

.

Taking into account the last two remarks, we can rewrite Eq. (A.12) in a more compactform:

L℘ = F⋆ , (A.13)

where the linear operator L and the right hand side F⋆ are defined as

L℘def:= ∇·

∇℘

H−

(∇℘·∇h

)∇h

HΥ

− 6℘

2H3

Υ − 3Υ

+ ∇·

[∇h

H2 Υ

]

,

F⋆ def:= g∆η + ∇·

Q∇h

Υ− Λ

−6Q

HΥ

+ 2(∇·u)2 − 2(u1λu2

θ − u1θ u2

λ

)− 2u2(u1

λ + u2θ

)cotθ + (u2)2 .

By expanding divergences of covariant vectors, we can express the operatorL(·) throughpartial derivatives:

L℘ ≡1

R2 sinθ

1

sinθ

[℘λ

H−

∇℘·∇h

HΥhλ

]

λ

+

[(℘θ

H−

∇℘·∇h

HΥhθ

)

sinθ

]

θ

− 6℘

2H3

Υ − 3Υ

+1

R2 sinθ

[1

sinθ

( hλ

H2 Υ

)

λ

+( hθ

H2 Υsinθ

)

θ

]

.


After multiplying both sides of Eq. (A.13) by R2 sinθ and switching to linear componentsof the velocity u and v, we obtain the following equation for ℘:[

1sinθ

℘λ

H−

∇℘·∇h

HΥhλ

]

λ

+

[℘θ

H−

∇℘·∇h

HΥhθ

sinθ

]

θ

− 6℘[

2H3

(Υ − 3)R2 sinθ

Υ+

( hλ

H2 Υsinθ

)

λ

+( hθ

H2Υsinθ

)

θ

]

= F , (A.14)

where the right hand side F is defined as

Fdef:=

[1

sinθ

gηλ +Q

Υhλ − Λ1

]

λ

+

[

gηθ +Q

Υhθ − Λ2

sinθ

]

θ

− R2 6Q

HΥsinθ +

2sinθ

uλ + (vsinθ)θ

2

− 2(uλvθ − vλuθ

)− 2(uv)λcotθ −

(v2 cosθ

)

θ.

The quantities Q , B and Λ are expressed through linear velocity components as

Q ≡(Λ − g∇η

)·∇h +

1R2 sinθ

u2

sinθhλλ + 2uvhλθ + v2hθθ sinθ

+ B ,

B ≡ htt + 2 u

Rsinθhλt +

v

Rhθ t

,

Λ1def:= −

(2uvcotθ + vR

)sinθ , Λ2

def:= u2 cotθ + uR,

Λ·∇h ≡Λ1hλ

R2 sin2 θ+

Λ2 hθ

R2 .

This concludes naturally the derivation of the elliptic equation (A.14) for the dispersivepart of the depth-integrated pressure ℘.

Remark A.1. Let us assume that the angular velocity vanishes, i.e. Ω ≡ 0. In a vicin-ity of a fixed admissible point

(λ⋆,θ⋆

)we introduce a non-degenerate transformation of

coordinates:xX

def:= R(λ − λ

⋆)sinθ⋆ , yXdef:= −R(θ − θ⋆),

along with corresponding velocity components:

uX ≡ ˙xX = Rλsinθ ≡ ςu, vX ≡ ˙yX = −R θ = −v,

where ςdef:= sinθ⋆

sinθ . By assuming that the considered neighbourhood is small in the latitude,

i.e. the quantity εdef:= |θ − θ⋆ | ≪ 1 is small. After transforming Eq. (A.14) to new

independent(xX,yX

)and new dependent

(uX,vX

)variables and neglecting the terms

of the order O(ε) and O(R−1) one can obtain the familiar to us elliptic equation for ℘ inthe globally plane case [46]. This fact is a further confirmation of the correctness of ourderivation procedure described above.


B Acronyms

In the text above the reader could encounter the following acronyms:

BBM BENJAMIN–BONA–MAHONY

FEM Finite Element Method

SOR Successive Over–Relaxation

NOAA National Oceanic and Atmospheric Administration

NSWE Nonlinear Shallow Water Equations

FNWD Fully Nonlinear Weakly Dispersive

WNWD Weakly Nonlinear Weakly Dispersive

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